WEBVTT
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π is the set of all integers from one to π.
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πΈ is the set of even numbers.
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π is the set of multiples of five.
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π is the set of prime numbers.
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Part a, explain why there is only one number in πΈ intersection π.
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Part b, given that π is a multiple of 10, how many numbers are in π intersection π?
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Circle your answer.
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Is it π divided by five minus one, π divided by five plus one, π minus five, or π divided by five?
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Letβs just remind ourselves of some of the notation that we need to know.
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This is the Greek letter π.
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And itβs usually used to denote the universal set.
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Thatβs all the objects weβre interested in.
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And here, those are all integers.
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Remember, those are whole numbers from one to π.
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So we could say that that could be one, two, three, four, all the way through to π.
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We then see that πΈ is the set of all even numbers.
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Remember, an even number is a multiple of two.
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And it will always end in two, four, six, eight, or zero.
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π is the set of multiples of five.
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And those are the numbers in the five times tables.
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Theyβre five, 10, 15, 20, and so on.
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And then, we have π as being the set of prime numbers.
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Remember, a prime number has exactly two factors, one and itself.
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The first five prime numbers are two, three, five, seven, and 11.
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Itβs important to remember that one is not a prime number.
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One actually only has one factor, one.
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So how does this help us for part a?
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Well, this symbol here that looks a little bit like the letter n means the intersection of two sets.
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When we think about that in terms of two circles on a Venn diagram, itβs the overlap.
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So weβre interested in the numbers that would appear in the overlap between the set of even numbers and the set of prime numbers.
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That is the set of numbers that are both even numbers and prime numbers.
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Two is the only even prime number.
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This means it is the only number which will appear in the intersection between sets πΈ and π.
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Remember, we said a prime number has exactly two factors.
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Two has two factors; itβs one and two.
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Any other even number will have a factor of two apart from one and itself.
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So thereβs no way that any other even number can have only two factors.
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And we can now see that there can only be one number in πΈ intersection π.
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Itβs the number two, since two is the only even prime number.
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Now, letβs consider part b.
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Weβre told that π is a multiple of 10.
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Remember, multiples of 10 are the numbers in the 10 times table: 10, 20, 30, and so on.
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And weβre looking to find how many numbers there are in the intersection of sets π and π.
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That means weβre interested in the multiples of five which are also integers from one to π.
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If π is equal to 10, then numbers that appear in this set are five and 10.
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If π is equal to 20, the multiples of five in this set are five, 10, 15, and 20.
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And if π is equal to 30, the multiples are five, 10, 15, 20, 25, and 30.
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That means that when π is equal to 10, there are two numbers in the set.
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When π is equal to 20, there are four numbers in the set.
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And when π is equal to 30, there are six numbers in the set.
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In fact, we can see that for every 10 numbers there are in π, there are two multiples of five.
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When π is a multiple of 10, then we can say that two out of every 10 of these numbers are multiples of five.
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Thatβs two-tenths of π.
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Remember, βofβ is commonly interchanged with the multiplication symbol.
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And we can simplify two-tenths to one-fifth.
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And we say that one-fifth of π are multiples of five when π is a multiple of 10.
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Another way of saying this is π over five.
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So our answer here is π over five.